03-11-2014, 11:35 AM
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1. Remainder Theorem
If a polynomial f(x) is divided by x - a, then the remainder, R = f(a).
2. Factor Theorem
If f(a) = 0, then (x - a) is a factor of f(x)
3. Solutions of EquationsTo solve the equation f(x) = 0, first factorize f(x) by the Factor Theorem
Eg . solve x[sup]3[/sup] - 2x[sup]2[/sup] - 5x + 6 = 0
Let f(x) = x[sup]3[/sup] - 2x[sup]2[/sup] - 5x + 6
Find a factor by trial and error
eg, found that (x - 1) is a factor
Divide f(x) by (x - 1) and we get
f(x) = (x - 1)(x[sup]2[/sup] - x - 6)
Factorize x[sup]2[/sup] - x - 6
---> f(x) = (x - 1)(x + 2)(x - 3)
so, (x - 1)(x + 2)(x - 3) = 0
so x = 1, -2, or 3Example Questions
Note: How to divide f(x) by (x + 1)
Note: How to factorize 4x[sup]2[/sup] -12x +9
Questions
Answers1a. -2, 3, 5
1b. 1, -2, -1/2
1c. 1/2, -1/2, 2/3
2. 2
3a. a = 3, b = 8
3b. (2x - 1)(x - 1)
4. -2
5. c = 0 , 3, -3
If a polynomial f(x) is divided by x - a, then the remainder, R = f(a).
2. Factor Theorem
If f(a) = 0, then (x - a) is a factor of f(x)
3. Solutions of EquationsTo solve the equation f(x) = 0, first factorize f(x) by the Factor Theorem
Eg . solve x[sup]3[/sup] - 2x[sup]2[/sup] - 5x + 6 = 0
Let f(x) = x[sup]3[/sup] - 2x[sup]2[/sup] - 5x + 6
Find a factor by trial and error
eg, found that (x - 1) is a factor
Divide f(x) by (x - 1) and we get
f(x) = (x - 1)(x[sup]2[/sup] - x - 6)
Factorize x[sup]2[/sup] - x - 6
---> f(x) = (x - 1)(x + 2)(x - 3)
so, (x - 1)(x + 2)(x - 3) = 0
so x = 1, -2, or 3Example Questions
Note: How to divide f(x) by (x + 1)
Note: How to factorize 4x[sup]2[/sup] -12x +9
Questions
Answers1a. -2, 3, 5
1b. 1, -2, -1/2
1c. 1/2, -1/2, 2/3
2. 2
3a. a = 3, b = 8
3b. (2x - 1)(x - 1)
4. -2
5. c = 0 , 3, -3